Hyponormal and strongly hyponormal matrices in inner product spaces

Linear Algebra and its Applications 433 (2010) 1055–1076

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Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa

Hyponormal and strongly hyponormal matrices in inner product spaces Mark-Alexander Henn a , Christian Mehl b , Carsten Trunk c ,∗ a b c

Physikalisch-Technische Bundesanstalt, Abbestr. 2-12, D-10587 Berlin, Germany Institute of Mathematics, MA 4-5, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany Department of Mathematics, Technische Universität Ilmenau, Postfach 100565, D-98648 Ilmenau, Germany

A R T I C L E

I N F O

Article history: Received 6 April 2009 Accepted 15 April 2010 Available online 23 June 2010 Submitted by C.K. Li AMS classification: Primary: 15A57 Secondary: 15A63

A B S T R A C T

Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on earlier works on normal matrices, the notions of hyponormal and strongly hyponormal matrices are introduced. A full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces. Finally, the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed. © 2010 Elsevier Inc. All rights reserved.

Keywords: Degenerate inner product Adjoint Linear relations H-Hyponormal Strongly H-hyponormal Invariant semidefinite subspace

1. Introduction We consider the space Cn equipped with an indefinite inner product [·, ·] that is not necessarily nondegenerate, i.e., there may exist vectors x ∈ Cn \ {0} such that [x, y] = 0 for all y ∈ Cn . In the case

∗ Corresponding author. E-mail addresses: [email protected] (M.-A. Henn), [email protected] (C. Mehl), carsten.trunk@ tu-ilmenau.de (C. Trunk).

0024-3795/$ - see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.04.050

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of a nondegenerate inner product matrix T [∗] satisfying

[·, ·], the adjoint of a matrix T with respect to [·, ·] is the unique

[x, Ty] = [T [∗] x, y] for all x, y ∈ Cn .

(1)

As usual, one defines H-selfadjoint, H-skewadjoint, H-unitary, and H-normal matrices, as matrices satisfying T [∗]

= T , T [∗] = −T , T [∗] = T −1 , and T [∗] T = TT [∗] ,

(2)

respectively. Introducing the Gram matrix H via

[x, y] = (Hx, y), where (·, ·) denotes the standard Euclidean scalar product, the adjoint can be expressed as T [∗]

= H −1 T ∗ H

and the identities in (2) reduce to HT

= T ∗ H, T ∗ H + HT = 0, T ∗ HT = H, HTH −1 T ∗ H = T ∗ HT ,

(3)

respectively. H-Selfadjoint, H-skewadjoint, H-unitary, and H-normal matrices have been studied extensively in the literature. Interest is motivated by various applications such as the theory of zones of stability for linear differential equations with periodic coefficients, see [8], the theory of algebraic Riccati equations, see [9], and the linear quadratic optimal control problems as in [17]. A concise overview of the theory of matrices in spaces with an indefinite inner product can be found in [3,4], see also [5] for H-normal matrices. An even more general class of matrices is the set of H-hyponormal matrices that are defined by analogy to the well-known class of hyponormal operators in Hilbert spaces via the condition H (T [∗] T

− TT [∗] )  0.

(4)

For negative definite H, the set of these matrices equals the set of H-normal matrices, but in the case that H is not definite, the set of H-hyponormal matrices is a proper superset of the set of H-normal matrices. H-hyponormal matrices were studied in detail in [13,14], where, in particular, extension results of invariant semidefinite subspaces to invariant maximal semidefinite subspaces were obtained. Spaces with a degenerate inner product, i.e., the Gram matrix H is singular, are less familiar, although this case does appear in applications, e.g., in the theory of operator pencils, cf. [10]. The main problem in this context is that there is no straightforward definition of an H-adjoint. Indeed, if H is noninvertible, the H-adjoint of a matrix T ∈ Cn×n need not exist. For example consider     1 0 1 1 H= , and T = . 0 0 0 1 Then a simple calculation shows that there is no matrix N

∈ C2×2 such that

(Hx, Ty) = [x, Ty] = [Nx, y]. In [15], H-selfadjoint, H-skewadjoint, and H-unitary matrices were defined by using the matrix identities from (3). The corresponding equation for H-normal matrices, however, requires an inverse of H. One way to modify this definition is the use of the well-known Moore–Penrose generalized inverse H † of H. In [11], a matrix T is called H-normal if HTH † T ∗ H

= T ∗ HT .

We will call such matrices Moore–Penrose H-normal matrices in this paper. In [16], a different definition of H-normal matrices in degenerate inner product spaces was used which is based on a generalization of the H-adjoint T [∗] of a matrix T for the case of singular H. This is obtained by dropping the assumption that the H-adjoint of a matrix has to be a matrix itself. Instead,

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the H-adjoint T [∗] is understood as a linear relation in Cn , i.e., a subspace of C2n . Clearly, every matrix T ∈ Cn×n can be interpreted as a linear relation in Cn by identification with its graph    x , x ∈ Cn ⊆ C2n . Γ (T ):= Tx If H

∈ Cn×n is invertible, then T [∗] , defined as in (1), coincides with the linear relation   y z



∈ C2n : [y, Tx] = [z, x] for all x ∈ Cn .

(5)

Hence, it is natural to define the adjoint T [∗] of T with respect to some degenerate inner product as the linear relation given in (5). This approach was used in [16] to generalize the notion of H-normal matrices to degenerate inner product spaces: a matrix T ∈ Cn×n is called H-normal if TT [∗] ⊆ T [∗] T . It was then shown in [16] that H-normal matrices T have the property that the kernel of H is T -invariant. This fact allowed the immediate generalization of extension results of invariant semidefinite subspaces from [13,14] to the degenerate case. However, the fact that the kernel of H is invariant is not needed in order to obtain results on the existence of invariant maximal semidefinite subspaces. Indeed, it was shown in [12] that the kernel of H need not be invariant for Moore–Penrose H-normal matrices, but it is always contained in an invariant H-neutral subspace. This property was used in the proof of the existence of invariant H-nonnegative subspaces for Moore–Penrose H-normal matrices in [12]. In this paper, we continue the work started in [16] by generalizing the concept of H-hyponormality to the case of degenerate inner product spaces. Our aim is to do this in such a way that the obtained set of matrices (i) contains the sets of H-normal and Moore–Penrose H-normal matrices; (ii) equals the set of H-normal matrices when H is negative semi-definite; (iii) guarantees that the kernel of H is always contained in an invariant H-neutral subspace. The latter condition will allow the generalization of existence results for invariant maximal H-nonpositive subspaces. After reviewing some basic results on linear relations in degenerate indefinite inner product spaces in Section 2, we introduce H-hyponormal matrices in Section 3. It turns out that this rather straightforward generalization of H-hyponormality is not satisfactory as the resulting matrices are too general. Therefore, the more restrictive concept of strong hyponormality is introduced in Section 4. In Section 5, we investigate the relation of H-hyponormal and strongly H-hyponormal matrices to Moore–Penrose H-normal matrices. In particular, we show that the set of Moore–Penrose H-normal matrices is a proper subset of the sets of strongly H-hyponormal and H-hyponormal matrices. Finally, we give sufficient conditions for the existence of invariant H-nonpositive subspaces for strongly H-hyponormal matrices in Section 6. 2. Preliminaries For the remainder of the paper let H ∈ Cn×n be a possibly singular Hermitian matrix and let [·, ·] denote the possibly degenerate inner product given by

[x, y]:=(x, Hy) for x, y ∈ Cn×n . If L

⊂ Cn is a subspace, the H-orthogonal companion of L (in Cn ) is defined by [⊥]

L

:={x ∈ Cn : [x, ] = 0 for all  ∈ L}.

The isotropic part of L is defined by ◦

L

:= L ∩ L[⊥] .

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The subspace L is called nondegenerate (or, more precisely, H-nondegenerate) if L◦ = {0}. If N ⊂ Cn×n is a subspace with N ⊂ L[⊥] we write N [⊥]L. If, in addition, N ∩ L = {0}, then by N []L we denote the direct H-orthogonal sum of N and L. A vector x ∈ Cn×n is called H-positive (H-negative, H-neutral) if [x, x] > 0 (resp. [x, x] < 0, [x, x] = 0), and H-nonnegative (H-nonpositive) if x is not H-negative (resp. not H-positive). A subspace L ⊂ Cn×n is called H-positive (H-negative, H-neutral, H-nonnegative, H-nonpositive) if all vectors in L \ {0} are H-positive (resp. H-negative, H-neutral, H-nonnegative, H-nonpositive). Observe that by this definition the zero space {0} is both H-positive and H-negative. The subspace L is called maximal H-nonpositive if it is H-nonpositive and if there is no nonpositive subspace L = / L containing L. For basic facts concerning the geometry in spaces with a degenerate inner product we refer to [1]. 2.1. Linear relations The proofs for the propositions and lemmas used in this section can be found, e.g., in [2,7,16]. A linear relation in Cn is a linear subspace of C2n . A matrix T ∈ Cn×n can be interpreted as a linear relation in Cn via its graph Γ (T ), where    x Γ (T ):= , x ∈ Cn . Tx Keeping this in mind, the following definitions are quite familiar. Definition 2.1. For linear relations S, T ⊆ C2n we define     x ∈ S , the domain of S ; dom S = x : y     0 mul S = y : ∈S , the multivalued part of S ; y      y x S −1 = : y ∈S , the inverse of S ; x        x x x S+T = : y ∈ S, z ∈ T , the sum of S, and T ; y+z and the product of S and T   x ST = : there exists a y z If dom S

  y

  x



∈ Cn with z ∈ S, y ∈ T .

= Cn , we say that S has full domain. In all the cases x, y, z are understood to be from Cn .

Note that a linear relation is always invertible in the above sense. We now give a more general definition of the H-adjoint of a linear relation T . This coincides with the linear relation in (5) when T is a matrix. Definition 2.2. Let [·, ·] denote the indefinite inner product induced by H, and let T be a linear relation in Cn . Then the linear relation      x y 2n ∈ C : [ y, w] = [z, x] for all ∈ T T [∗] = w z is called the H-adjoint of T . The next proposition which is a summary of Proposition 2.2 and Lemma 2.4 of [16] contains some basic properties of the H-adjoint. Proposition 2.3. Let S, T (i) S

⊆ C2n be linear relations. Then

⊆ T implies T [∗] ⊆ S[∗] ;

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

(ii) (iii) (iv) (v) If T

S [∗] + T [∗] ⊆ (S + T )[∗] ; T [∗] S [∗] ⊆ (ST )[∗] ; mul T [∗] = (dom T )[⊥] ; if T is a matrix, then mul T [∗] (T [∗] )[∗] = T + (ker H × ker H ).

∈ Cn×n is a matrix, then T [∗]

=

  y z

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= ker H ;



∈ C2n : T ∗ Hy = Hz .

(6)

In particular, T [∗] is a matrix if and only if H is invertible. Note that we can always find a basis of Cn such that the matrices H and T have the forms     H1 T 0 T2 H= ∈ Cn× n , and T = 1 0 0 T3 T4

(7)

where H1 , T1 ∈ Cm×m , m  n, and H1 is invertible. Using this decomposition, T can be written as ⎧⎛ ⎧⎛ ⎫ ⎞ ⎞⎫ x1 x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎨ ⎬ ⎜ ⎟ ⎟⎬ x2 x2 m n −m ⎜ ⎜ ⎟ ⎟ T = ⎝ : x ∈ C = , x ∈ C 1 2 ⎝T x + T2 x2 ⎠⎪ . T x + T2 x2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 1 ⎩ 1 1 ⎭ ⎭ T3 x 1 + T 4 x2 T3 x 1 + T 4 x 2 We will use the short second notion when it is clear from the context how the matrices T and H are decomposed. Proposition 2.4 ([16]). Let T T [∗]

∈ Cn×n be a matrix. Then

= H −1 T ∗ H,

where H −1 denotes the inverse of H in the sense of linear relations. Furthermore, if H and T have the forms as in (7), then ⎧⎛ ⎫ ⎞ y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ y2 ⎟ ⎬ ⎜ ⎟ [∗] ∗ T = ⎪⎜ [∗]H1 ⎟ : T2 H1 y1 = 0⎪ , (8) ⎝T y1 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎭ z2 [∗]H

[∗]H1

where T1 1 denotes the adjoint with respect to the invertible matrix H1 , i.e., T1 particular, dom T [∗] = Cn if and only if T2 = 0. [∗]

We will suppress the subscript H1 , writing T1 H1 induces the indefinite inner product.

[∗]H1

instead of T1

= H1−1 T1∗ H1 . In

when it is clear from the context that

2.2. H-Symmetric linear relations As usual, a linear relation T ⊆ C2n is called H-symmetric if T ⊆ T [∗] , see, e.g., [19]. The notion of H-symmetric linear relations will be needed when we introduce the class of H-hyponormal matrices in Section 3. We therefore collect some of the basic properties of H-symmetric matrices and linear relations from [16, Propositions 3.3 and 3.4, Corollary 3.5]. Proposition 2.5. Let T

∈ Cn×n be a matrix. Then the following statements are equivalent:

(i) T is H-symmetric, i.e., T (ii) T ∗ H = HT ;

⊆ T [∗] ;

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(iii) T [∗] (iv) T [∗]

= (T [∗] )[∗] ; = T + (ker H × ker H ).

If one of the conditions is satisfied, then ker H is T -invariant. In particular, if H and T have the forms as in (7) then T is H-symmetric if and only if T1 is H1 -selfadjoint and T2 = 0. Recall from Proposition 2.3 that in the case of a singular H the relation T [∗] is never a matrix. Hence, a matrix with T = T [∗] does not exists. Therefore, in view of Theorem 2.5(iii), H-symmetry can be considered as a generalization of the notion of H-selfadjointness for the case of a singular H. The following proposition will be an important tool in Section 3. Proposition 2.6. Let T

⊂ C2n be a linear relation:

(1) TT [∗] and T [∗] T are H-symmetric, i.e., TT [∗] (2) If T

⊆ (TT [∗] )[∗] and T [∗] T ⊆ (T [∗] T )[∗] .

n ×n

∈C

is a matrix, then the following assertions are equivalent:

(i) The domain of the linear relation T [∗] T is Cn , (ii) T [∗] T = (T [∗] T )[∗] . In particular, if T and H are in in the form (7), then (i) and (ii) are equivalent to (iii) T2∗ H1 T1

= 0 and T2∗ H1 T2 = 0.

Proof. (1) Proposition 2.3(v) implies T TT

[∗]

⊆ (T

[∗] [∗] [∗]

) T

⊆ (TT

⊆ (T [∗] )[∗] . Hence, by Proposition 2.3(iii), we have that

[∗] [∗]

) .

Analogously, we obtain T [∗] T

⊆ T [∗] (T [∗] )[∗] ⊆ (T [∗] T )[∗] . (2) Assume now that T ∈ Cn×n is a matrix. Without loss of generality, we may assume that T and H are in the form (7). Then Proposition 2.4 implies ⎧⎛ ⎞ ⎪ y1 ⎪ ⎪⎜ ⎨ ⎟ y2 ⎜ ⎟ T [∗] T = ⎜ [∗] ⎟ : T2∗ H1 T1 y1 [∗] ⎪ ⎝T 1 T 1 y 1 + T 1 T 2 y 2 ⎠ ⎪ ⎪ ⎩ z

⎫ ⎪ ⎪ ⎪ ⎬

+ T2∗ H1 T2 y2 = 0 . ⎪ ⎪ ⎪

2

(9)

⎭

Hence, (iii) and (i) are equivalent. If (iii) holds, we have 0

= (T2∗ H1 T1 )∗ = T1∗ H1 T2 and 0 = H1−1 T1∗ H1 T2 = T1[∗] T2 .

By a simple calculation together with (9) we obtain ⎧⎛ ⎞⎫ ⎪ ⎪ y1 ⎪ ⎪ ⎪ ⎨⎜ y ⎟⎪ ⎬ 2 ⎜ ⎟ [∗] T T = ⎜ [∗] ⎟ = (T [∗] T )[∗] ⎪ ⎝T T1 y1 ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 z ⎭ 2

and (ii) follows. For the remainder of the proof we assume that (ii) holds. If there exists y2 with T2∗ H1 T2 y2   0 ∈ / dom T [∗] T . But by (9), y2 ⎛ ⎞ 0 ⎜y2 ⎟ ⎜ ⎟ ∈ (T [∗] T )[∗] ⎝0⎠ 0

= / 0, then,

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a contradiction to (ii). Therefore, (ii) implies T2∗ H1 T2 y2 = 0 and we have by (9) that ⎧⎛ ⎫ ⎞ ⎪ ⎪ y1 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎬ ⎟ y2 ⎜ ⎟ [∗] ∗ : T H T y = 0 . ⎟ T T = ⎜ [∗] [∗] 1 1 1 2 ⎪ ⎪ ⎝T T 1 y 1 + T 1 T 2 y 2 ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎭ z2 Now let w1 ∈ (ker T2∗ H1 T1 )[⊥]H1 , that is [w1 , y1 ] ⎛ ⎞ 0 ⎜0⎟ ⎜ ⎟ ∈ (T [∗] T )[∗] = T [∗] T , ⎝w1 ⎠ 0 hence w1

= 0 for all y1 satisfying T2∗ H1 T1 y1 = 0. Then

= 0 and T2∗ H1 T1 y1 = 0 follows. Thus (ii) implies (iii). 

Remark 2.7. We mention that TT [∗] has full domain if and only if T [∗] has full domain, which is equivalent to the fact that T2 = 0, cf. Proposition 2.4. However, a similar statement as the equivalence of (i) and (ii) in Proposition 2.6, part 2, does not hold for TT [∗] . As an example consider the matrix T = 0. Then TT [∗] is the zero matrix but mul(TT [∗] )[∗] = ker H, see Proposition 2.3. Hence TT [∗] has full domain but TT [∗] = / (TT [∗] )[∗] . 2.3. H-Normal matrices In the case of a singular H, where the matrices H and T are given in the forms as in (7), it is easily checked that ⎧⎛ ⎫ ⎞ y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎬ y2 ⎜ ⎟ [∗] ∗ = ⎪⎜T T [∗] y + T y ⎟ : T2 H1 y1 = 0 . (10) TT ⎪ ⎝ ⎠ 1 1 1 2 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ [∗] T3 T1 y1 + T4 z 2 Comparing (9) and (10) one can easily see that in the case of an H-symmetric T we obtain only that TT [∗] ⊆ T [∗] T and the inclusion T [∗] T ⊆ TT [∗] is only satisfied in the case that T4 is invertible. Therefore, H-normal matrices were defined in [16] by the inclusion TT [∗]

⊆ T [∗] T

(11)

rather than by the identity TT [∗]

=

T [∗] T , because otherwise there would exist T -symmetric matrices

that are not H-normal. It was then shown in [16, Proposition 4.2] that for H-normal matrices ker H is T-invariant and that, if T and H are given in the forms (7), T is H-normal if and only if T1 is H1 -normal and T2 = 0. 3. H-Hyponormal matrices If the Hermitian matrix H satisfies H (T [∗] T

∈ Cn×n is invertible, then an H-hyponormal matrix T by definition

− TT [∗] )  0,

i.e., T [∗] T − TT [∗] is H-nonnegative. Such matrices were discussed, e.g., in [13,14]. A generalization of this definition to the case of singular H requires the concept of H-nonnegativity for linear relations. Definition 3.1. A linear relation S and if   x [y, x]  0 for all y ∈ S.

⊆ C2n is called H-nonnegative if S is H-symmetric (i.e. S ⊆ S[∗] ) (12)

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Analogously the notions of H-nonpositivity, H-positivity and H-negativity of a linear relation are defined. The following lemma yields a base for the definition of H-hyponormal matrices. Lemma 3.2. Let T

⊆ C2n be a linear relation. Then the relation T [∗] T − TT [∗] is H-symmetric.

Proof. Propositions 2.6 and 2.3(ii) imply T [∗] T

− TT [∗] ⊆ (T [∗] T )[∗] − (TT [∗] )[∗] ⊆ (T [∗] T − TT [∗] )[∗] .



In the following, we will give conditions which ensure the H-nonnegativity of T [∗] T case that T is a matrix, the following characterization holds.

− TT [∗] . In the

Proposition 3.3. Let T ∈ Cn×n be a matrix and let T and H be in the forms (7). Then T [∗] T H-nonnegative if and only if

− TT [∗] is

[∗]

− T1 T1[∗] )y1  y2∗ T2∗ H1 T2 y2 for all y1 , y2 satisfying T2∗ H1 y1 = 0 and T2∗ H1 (T1 y1 + T2 y2 ) = 0. y1∗ H1 (T1 T1

Proof. The linear relation T [∗] T − TT [∗] is H-symmetric by Lemma 3.2. Then we obtain from (9) and (10) that T [∗] T − TT [∗] equals ⎧⎛ ⎫ ⎞ y1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎨⎜ ⎬ y2 T2∗ H1 y1 = 0; ⎟ ⎜ [∗] [∗] . ⎟: ∗ ⎜ [∗] ( T T − T T H ( T T ) y + T T y + T y ) = 0 y − T y ⎪ ⎪ ⎝ 1 1 1 1 1 2 2 2 2 2 2⎠ 1 ⎪ ⎪ 2 1 1 1 ⎪ ⎪ ⎩ ⎭ [∗] w 2 − T3 T1 y1 − T4 z 2 Thus, T [∗] T

− TT [∗] is H-nonnegative if and only if [∗]

− T1 T1[∗] )y1 + y1∗ H1 T1[∗] T2 y2 − y1∗ H1 T2 y2  0 (13) for all y1 , y2 that satisfy T2∗ H1 y1 = 0 and T2∗ H1 (T1 y1 + T2 y2 ) = 0. The restrictions on y1 and y2 imply y1∗ H1 (T1 T1 y1∗ H1 T2 y2

= 0 and y1∗ H1 T1[∗] T2 y2 = y1∗ T1∗ H1 T2 y2 = −y2∗ T2∗ H1 T2 y2 . [∗]

Thus, (13) reduces to y1∗ H1 (T1 T1

− T1 T1[∗] )y1  y2∗ T2∗ H1 T2 y2 . 

At this point, one could define H-hyponormal matrices as matrices T for which the linear relation T [∗] T − TT [∗] is H-nonnegative. However, this would not be satisfactory, because the class of matrices obtained in this way is too general. In particular, an important property of H-hyponormal matrices would be lost. It is well known that if H is negative definite then H-hyponormality implies H-normality, see, e.g., [6]. However, if we relax the condition of H being negative definite to H being negative semidefinite, then H-nonnegativity of T [∗] T − TT [∗] is no longer sufficient for H-normality as the following example shows. Example 3.4. Let r, m, n be such that r

H

=



−Im 0



0 , 0

T

=



T1 T3

T2 T4

< m, r + m < n and let 

⎡

=

T11 ⎢T13 ⎢ ⎣T31 T33

T12 T14 T32 T34

Ir 0 T41 T43

⎤ 0 0 ⎥ ⎥, T42 ⎦ T44

where H ∈ Cn×n , T11 ∈ Cr ×r , T41 ∈ Cr ×r , T44 ∈ C(n−r −m)×(n−r −m) , and where T14 ∈ C(m−r )×(m−r ) ∗ T = T T ∗ . Moreover, let is normal with respect to the standard Euclidean product, i.e., T14 14 14 14 ⎛ ⎞ y11   ⎜y12 ⎟ y1 ⎟ y= =⎜ ⎝y21 ⎠ y2 y22

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

be partitioned conformably with T . Then by (9) and (10), y is in the domain of T [∗] T only if T2∗ H1 y1

− TT [∗] if and

= 0 and T2∗ H1 (T1 y1 + T2 y2 ) = 0.

The first identity implies that     Ir y11 0 −Ir 0 0 0 0 −Im−r y12 Thus, if y is in the domain of T [∗] T T2∗ H1 T1 y1 that is

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=



−y11 0



(14)

= 0.

− TT [∗] , then y11 = 0. The second identity in (14) implies

= −T2∗ H1 T2 y2 ,



−T11 y11 − T12 y12



0 therefore 

−T11 y11 − T12 y12 0



(15) 

 0 −Ir 0 0

I

= − 0r

=



y21 0

Hence y is in the domain of T [∗] T ⎛ ⎞ 0   ⎜ y12 ⎟ y1 ⎟ y= =⎜ ⎝−T12 y12 ⎠ . y2 y22



0 −Im−r



Ir 0

  0 y21 , 0 y22

.

− TT [∗] if and only if it has the form: (16)

Due to the normality of T14 and because of (15), we obtain that [∗]

y1∗ H1 (T1 T1

∗ ∗ ∗ ∗ ∗ (T12 − T1 T1[∗] )y1 = −y12 T12 + T14 T14 − T13 T13 − T14 T14 )y12 ∗ ∗ ∗ ∗ ∗ = −y12 (T12 T12 − T13 T13 )y12  −y12 T12 T12 y12 = y2∗ T2∗ H1 T2 y2 .

Thus by Proposition 3.3, T [∗] T − TT [∗] is H-nonnegative. However, if r because the kernel of H is not T-invariant, see [16, Proposition 4.2].

> 0 then T is not H-normal,

Moreover, in the above Example 3.4 we have T2∗ H1 T2 = / 0 and by [16, Relation (4.6) in the proof of Proposition 4.6] the matrix T is not Moore–Penrose H-normal. Recall that a matrix is called Moore– Penrose H-normal if HTH † T ∗ H

= T ∗ HT ,

where H † is the Moore–Penrose generalized inverse of H, see also Section 5 below. Example 3.4 shows that the reason for the failure of the desired property is the domain of T [∗] T − [∗] TT which is too small, in general. One way to circumvent this problem is to require that the linear relation TT [∗] has full domain. From (10) it follows that this is equivalent to T2 = 0 (if H and T are assumed to be in the form (7)) and thus, we then also have that T [∗] T and hence T [∗] T − TT [∗] have full domain. However, this condition is rather restrictive as T2 = 0 implies that the kernel of H is T invariant. Consequently, the set of matrices obtained in this way does not contain all Moore–Penrose H-normal matrices (we refer to [12, Example 6.1] for a Moore–Penrose H-normal matrix such that ker H is not invariant). Fortunately, it turns out that it is enough to require that T [∗] T has full domain in order to guarantee that the domain of T [∗] T − TT [∗] is sufficiently large so that H-nonnegativity of T [∗] T − TT [∗] implies H-normality in the case of negative semidefinite H. This motivates the following definition of H-hyponormality.

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M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

Definition 3.5. A linear relation T TT [∗] is H-nonnegative.

⊆ C2n is called H-hyponormal if T [∗] T has full domain and if T [∗] T −

From Propositions 3.3 and 2.6, we immediately obtain the following characterization of H-hyponormal matrices. Proposition 3.6. Let T ∈ Cn×n be a matrix and let T and H be in the forms (7). Then T is H-hyponormal if and only if T [∗] T has full domain and [∗]

− T1 T1[∗] )y1  0 for all y1 satisfying T2∗ H1 y1 = 0. y1∗ H1 (T1 T1

As a corollary, we obtain the desired property that H-hyponormality is equivalent to H-normality for negative semidefinite matrices H.

∈ Cn×n be negative semidefinite and let T ∈ Cn×n be an H-hyponormal matrix. Then

Corollary 3.7. Let H T is H-normal.

Proof. Without loss of generality let T and H be in the forms (7), where H1 = −Im . By Proposition 2.6, we have −T2∗ T2 = T2∗ H1 T2 = 0 which implies that T2 = 0. By Proposition 3.6, we then obtain that

−y1∗ (T1∗ T1 − T1 T1∗ )y1  0 for all y1 ∈ Cm , that is, T1∗ T1 − T1 T1∗ is negative semidefinite. Thus, all eigenvalues of T1∗ T1 − T1 T1∗ are smaller or equal to zero, and as tr(T1∗ T1 − T1 T1∗ ) = 0 it follows that T1∗ T1 − T1 T1∗ = 0 and hence T1 is normal. From Proposition [16, Proposition 4.2] we then get that T is H-normal.  In [16, Proposition 4.2], it was shown that the kernel of H is always T -invariant if T is H-normal, and it was shown in [12] that the kernel of H is always contained in a T -invariant H-neutral subspace if T is Moore–Penrose H-normal. Unfortunately, this is no longer true for H-hyponormal matrices as the following example shows. Example 3.8. Let

H

=



H1 0



0 0

⎡

=

−1 ⎢ 0 ⎢ ⎣ 0 0

0 1 0 0

Then one computes that T2∗ H1 T2 Moreover, [∗]

T1

⎡

=

0 ⎣0 0 [∗]

−1 0 0

⎤ 0 0⎦ 0

=

0 0 1 0

⎤ 0 0⎥ ⎥, 0⎦

T

=



T1 0

T2 0



⎡

=

0

0 and T2∗ H1 T1

=

and H1 (T1 T1

0 0 0 0

0 0 0 0

⎤ 1 0⎥ ⎥. 1⎦ 0

0 and, by Proposition 2.6, T [∗] T has full domain. ⎡

[∗]

0 ⎢1 ⎢ ⎣0 0

[∗]

− T1 T1 ) =

1 ⎣0 0

0 1 0

⎤ 0 0⎦ 0

[∗]

hence y∗ H1 (T1 T1 − T1 T1 )y  0 for all y ∈ C3 and, by Proposition 3.6, we obtain that T is H-hyponormal. However, note that ker H = span(e4 ) and that U := span(e4 , e1 + e3 , e2 ) is the smallest T -invariant subspace containing ker H. (Here, ei denotes the i-th standard basis vector.) Obviously, U is not H-neutral, as e2∗ He2 = 1. 4. Strongly H-hyponormal matrices We have seen in the previous section that H-hyponormal matrices are too general. We will therefore define a new class of matrices which is properly contained in the set of H-hyponormal matrices and small enough to ensure that the kernel of H is contained in an invariant H-neutral subspace.

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

1065

⊆ C2n be a linear relation:

Definition 4.1. Let T

(1) T is called strongly H-hyponormal of degree k ∈ N if T is H-hyponormal and if (T [∗] )i T i has full domain for all i = 1, 2, . . . , k. (2) T is called strongly H-hyponormal if T is strongly H-hyponormal of degree k for all k ∈ N. We start with two examples which show that the class of strongly H-hyponormal matrices neither coincides with the class of H-normal matrices nor with the class of H-hyponormal matrices. Example 4.2. Let H

=



Then T2∗ H1 T2 [∗]

T1

⎡



H1 0

0 0

−1 ⎣ 0 0

=

0 1 0

⎤ 0 0⎦ , 0

T

=



T1 0

T2 0



⎡

=

1 ⎣1 0

0 0 0

⎤ 1 1⎦ . 0

= 0 and T2∗ H1 T1 = 0, that is, T [∗] T has full domain. Moreover,





−1

1

= 0 [∗]

Hence H1 (T1 T1

0



[∗]

1

− T1 T1[∗] ) = −1

and H1 (T1 T1



−1 1

.

− T1 T1[∗] ) is positive semidefinite, i.e.,

[∗]

y1∗ H1 (T1 T1

− T1 T1[∗] )y1  0 = y2∗ T2∗ H1 T2 y2

for all y1 , y2 . Thus, T is H-hyponormal by Proposition 3.3. Moreover, T is also strongly H-hyponormal, because T is idempotent and, by Proposition 2.4, ⎧⎛ ⎞ ⎫ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎜ α ⎟ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎟ ⎨⎜ ⎟ ⎬ β [∗] ⎟ : α, β , γ ∈ C . T = ⎪⎜ ⎜0⎟ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎝0⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

γ

Hence, (T [∗] )k

= T [∗] and T k = T , k ∈ N, so that (T [∗] )k T k = T [∗] T has full domain for all k ∈ N. In

particular, T is an example for a matrix that is strongly H-hyponormal, but not H-normal, because T2 = / 0, i.e., ker H is not T -invariant, cf. [16, Proposition 4.2]. Example 4.3. The matrix given in Example 3.8 is not strongly H-hyponormal. Indeed, by Proposition 2.4, T [∗]

= {(y1 , y2 , y1 , y4 , −y2 , 0, 0, z4 )T : y1 , y2 , y4 , z4 ∈ C}

and one easily checks that ⎧⎛ ⎞ y1 ⎪ ⎪ ⎨⎜ ⎟ 0⎟ [∗] 2 dom(T ) = ⎜ ⎝y1 ⎠ : y1 , y4 ⎪ ⎪ ⎩ y4 and dom(T [∗] )2 T 2

=

⎫ ⎪ ⎪ ⎬

∈C

⎪ ⎪ ⎭

⎧⎛ ⎞ y1 ⎪ ⎪ ⎨⎜ ⎟ ⎜y2 ⎟ : y1 , y2 , y3 ⎝y ⎠ ⎪ ⎪ ⎩ 3 0

⎫ ⎪ ⎪ ⎬

∈ C⎪ . ⎪ ⎭

That is, T is H-hyponormal (see Example 3.8) but not strongly H-hyponormal.

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M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

In the following proposition, we characterize the property that (T [∗] )i T i has full domain in terms of T1 , T2 , and H1 when T and H are in the forms (7). We will use this characterization in the proof of our main result of this section, Theorem 4.5 below. Proposition 4.4. Let T ⊆ C2n be a linear relation. Then (T [∗] )k T k and T k (T [∗] )k are H-symmetric for all k ∈ N. In particular, if T ∈ Cn×n is a matrix and if T and H are in the forms (7), then the following assertions are equivalent: (1) (T [∗] )i T i has full domain for 1  i  k; [∗] i−1 i−1 ) T T

(2) T2∗ H1 (T1

1

1

= 0 and T2∗ H1 (T1[∗] )i−1 T1i−1 T2 = 0 for 1  i  k.

Proof. We first show that T i (T [∗] )i is H-symmetric, that is T i (T [∗] )i

⊆ (T i (T [∗] )i )[∗] .

By Proposition 2.3(iii) we find that (T [∗] )i T i (T [∗] )i

⊆ (T i )[∗] and therefore

⊆ T i (T i )[∗] .

(17)

From Propositions 2.3(v) and 2.3(iii), (i) it follows that Ti Hence T i

⊆ ((T i )[∗] )[∗] and ((T i )[∗] )[∗] ⊆ ((T [∗] )i )[∗] . ⊆ ((T [∗] )i )[∗] , and thus

T i (T i )[∗]

⊆ ((T [∗] )i )[∗] (T i )[∗] .

(18)

Putting together (17), (18), and using Proposition 2.3(iii), we obtain that T i (T [∗] )i

⊆ (T i (T [∗] )i )[∗] ,

thus T i (T [∗] )i is H-symmetric. A similar argumentation shows that (T [∗] )i T i is also H-symmetric. We next show by induction on k that   k −1 T1k + Bk T1 T2 + Ck k T = , (19)

∗

∗

where Bk

=

k  i =2

k −i

T1

(k )

T2 D i

and Ck

=

k  i =2

k −i

T1

(k ) T2  Di

(k ) (k ) for some matrices Di ∈ C(n−m)×m and  Di ∈ C(n−m)×(n−m) . For the case k = 1 there is nothing to show as B1 = 0 and C1 = 0 by the definition of the empty sum. If k  2 then    k −1 T2 T k + Bk T1 T2 + Ck T1 T k +1 = 1 T4 T3 ∗ ∗   k +1 k −1 T1 + Bk T1 + T1 T2 T3 + Ck T3 T1k T2 + Bk T2 + T1k−1 T2 T4 + Ck T4

=

∗

.

∗

By the induction hypothesis, we find that Bk T1

+ T1k−1 T2 T3 + Ck T3 =

k  i =2

k −i

T1

(k)

T2 D i T1

+ T1k−1 T2 T3 +

k  i =2

(k )

k −i

D i T3 T2 

k −i

(k) D i T4 . T2 

T1

and Bk T2

+ T1k−1 T2 T4 + Ck T4 =

k  i =2

k −i

T1

(k )

T2 D i T2

+ T1k−1 T2 T4 +

k  i =2

T1

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

1067

Thus, by setting (k + 1 )

D2

(k + 1 )

:= T3 , Di

(k )

(k )

Di−1 T3 for i = 3, . . . , k + 1 := Di−1 T1 + 

and (k + 1 ) (k + 1 ) (k )  := T4 ,  Di := Di−1 T2 D2 (k + 1 )

we obtain that Di Bk+1

=

we have T

k +1

k +1 i =2



=

(k + 1 )

∈ C(n−m)×n and  Di

k +1 −i

T1

k +1

T1

(k )

Di−1 T4 for i = 3, . . . , k + 1, +

(k + 1 )

T2 D i

+ B k +1 ∗

and Ck+1

T1k T2

+ C k +1 ∗

∈ C(n−m)×(n−m) , and therefore by setting =

k +1 i =2

k + 1 −i

T1

(k + 1 ) T2  Di ,

(20)



as desired. We now prove the equivalence of (1) and (2) by induction on k. The case k = 1 is covered by Proposition 2.6. If k  2, note that for j = 1, . . . , k + 1 we obtain using Proposition 2.4 that ⎫ ⎧⎛ ⎞ y1   ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y [∗] ⎪ y2  ⎟ ⎬ ⎨⎜ T2∗ H1 (T1 )s Wk+1 1 = 0; ⎪ ⎜ ⎟ [∗] j k+1 y ⎜ ⎟ 2 y1 ⎟ : , (21) (T ) T = ⎪⎜ [∗] j ⎪ ( T ) W k +1 ⎪ ⎪ s = 0, . . . , j − 1 ⎪ ⎪⎝ 1 y2 ⎠ ⎪ ⎪ ⎭ ⎩ z2     y y where Wk+1 1 is the first component of the vector T k+1 1 , i.e., y2 y2   y k +1 Wk+1 1 = (T1 + Bk+1 )y1 + (T1k T2 + Ck+1 )y2 . y2 Now assume that either (1) or (2) holds for 1  i  k + 1. Using the equivalence of conditions postulated in the induction hypothesis, we have in either case that (T [∗] )i T i has full domain for 1  i  k and that [∗] i−1 i−1

T2∗ H1 (T1

)

T1

T1

= 0, T2∗ H1 (T1[∗] )i−1 T1i−1 T2 = 0 for 1  i  k.

(22)

Thus it remains to show that the following assertions are equivalent: (a) (T [∗] )k+1 T k+1 has full domain; [∗] k k ) T1 T1

(b) T2∗ H1 (T1

= 0 and T2∗ H1 (T1[∗] )k T1k T2 = 0.

It follows from (21) that the relation (T [∗] )k+1 T k+1 has full domain if and only if (T [∗] )j T k+1 has full domain for all j = 1, . . . , k + 1. We will first that (22) implies that (T [∗] )j T k+1 has full domain  show  y [∗] 1 for j = 1, . . . , k that is T2∗ H1 (T1 )j−1 Wk+1 = 0 for all y1 , y2 and all j = 1, . . . , k. This means that y2 we have to show the following: [∗] j−1 k+1 ) T y

(i) T2∗ H1 (T1

= 0; = 0; = 0; 2 1 1 1 2 2 [∗] j−1 ∗ (iv) T2 H1 (T1 ) Ck+1 y2 = 0. 1

1

[∗] (ii) T2∗ H1 (T1 )j−1 Bk+1 y1 [∗] ∗ (iii) T H (T )j−1 T k T y

We easily obtain (i) and (iii) as [∗] j−1 k+1

T2∗ H1 (T1

)

T1

y1

j −1

k −j +1

= T2∗ H1 (T1[∗] )j−1 T1 T1 T1 

!

=0

"

y1

=0

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M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

and [∗] j−1 k

T2∗ H1 (T1

Moreover, T ∗ H 2

)

1

T1 T2 y2

(T [∗] )j−1 B 1

j −1

k −j

= T2∗ H1 (T1[∗] )j−1 T1 T1 T1 T2 y2 = 0. 

k +1

!

=0

"

= 0, because for i = 2, . . . , k + 1 we have

(k + 1 ) (k + 1 ) [∗] j−1 k+1−i = T2∗ (T1∗ )j−1 ((T1[∗] )∗ )k+1−i H1 T2 Di ) T1 T2 D i j−3−(k−i) [∗] k+1−i k+1−i ∗

T2∗ H1 (T1

= (T2 H1 (T1 ) 

for the case k + 1 − i < j − 1. Furthermore, for the case k + 1 − i [∗] j−1 k+1−i

T2∗ H1 (T1

)

T1

and for the case j − 1 ∗

)

=0

(k + 1 )

T2 )∗ Di

=0

= j − 1 we have j −1

(k + 1 )

= T2∗ H1 (T1[∗] )j−1 T1 T2 Di 

!

"

=0

=0


[∗] j−1 k+1−i

T2 H1 (T1

(k + 1 )

T2 D i

T1 T1 "

T1

!

T1

(k + 1 )

T2 D i

k +1 −i −j

j −1

= T2∗ H1 (T1[∗] )j−1 T1 T1 T1 

!

"

=0

(k + 1 )

T2 D i

=0

and from the definition of Bk+1 in (20) we conclude (ii). [∗] j−1 ) C

Moreover, T2∗ H1 (T1

k +1

= 0, because for i = 2, . . . , k + 1 we have

(k + 1 ) [∗] j−1 k+1−i (k+1) ) T1 T2 D i = T2∗ (T1∗ )j−1 ((T1[∗] )∗ )k+1−i H1 T2  Di

T2∗ H1 (T1

j−3−(k−i)

= (T2∗ H1 (T1[∗] )k+1−i T1k+1−i T1 T1 

for the case k + 1 − i < j − 1. Furthermore, for the case k + 1 − i [∗] j−1 k+1−i

T2∗ H1 (T1

)

T1

and for the case j − 1

)

T2 D i

"

=0

(k + 1 )

T2 )∗  Di

=0

= j − 1 we have j −1

(k+1)

Di = T2∗ H1 (T1[∗] )j−1 T1 T2  

!

"

=0

=0

< k + 1 − i we have

[∗] j−1 k+1−i

T2∗ H1 (T1

(k+1)

!

T1

(k + 1 ) Di T2 

j −1

k +1 −i −j

= T2∗ H1 (T1[∗] )j−1 T1 T1 T1 

!

=0

"

(k + 1 ) Di T2 

=0

and from the definition of Ck+1 in (20) we conclude (iv). Next, we will show that (22) also implies that [∗] k ) B k +1 y 1 [∗] (vi) T2∗ H1 (T1 )k Ck+1 y2

(v) T2∗ H1 (T1

= 0; = 0.

[∗] k ) B k +1 y 1

We show T2∗ H1 (T1

= 0. For i = 2, . . . , k + 1 we have

(k + 1 ) (k + 1 ) [∗] k + 1 −i T2 D i = T2∗ (T1∗ )k ((T1[∗] )∗ )k+1−i H1 T2 Di T2∗ H1 (T1 )k T1 k−2−(k−i)

= (T2∗ H1 (T1[∗] )k+1−i T1k+1−i T1 T1 

!

=0

"

(k + 1 )

T2 )∗ Di

= 0.

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

[∗] k )C

Moreover, T2∗ H1 (T1

k +1 y 2

1069

= 0, because for i = 2, . . . , k + 1 we have

(k + 1 ) [∗] k k+1−i (k+1) ) T1 T2 D i = T2∗ (T1∗ )k ((T1[∗] )∗ )k+1−i H1 T2  Di j−2−(k−i) [∗] k+1−i k+1−i ∗

T2∗ H1 (T1

= (T2 H1 (T1 ) 

!

T1

=0

T1 T1 "

(k + 1 )

T2 )∗  Di

= 0.

Thus, by (20), (v) and (vi) hold. We now have all ingredients to prove the equivalence of (a) and (b) under our induction hypothesis (22):

(T [∗] )k+1 T k+1 has full domain and (22)

$ % [∗] k +1 T2∗ H1 (T1 )s (T1 + Bk+1 )y1 + (T1k T2 + Ck+1 )y2 = 0, for all s = 0, . . . , k and all y1 , y2 , and (22) $ % # [∗] (i)–(iv) T2∗ H1 (T1 )k (T1k+1 + Bk+1 )y1 + (T1k T2 + Ck+1 )y2 = 0, ⇐⇒ for all y1 , y2 , and (22) (21)

#

⇐⇒

(v),(vi) ∗

⇐⇒ T2 H1 (T1[∗] )k (T1k+1 y1 + T1k T2 y2 ) = 0, for all y1 , y2 , and (22) #

⇐⇒

[∗] i−1 i−1 ) T T

T2∗ H1 (T1

for i

1

1

= 1, . . . , k + 1.

This concludes the proof.

= 0 and T2∗ H1 (T1[∗] )i−1 T1i−1 T2 = 0



Fortunately, it is not necessary to verify that (T [∗] )k T k has full domain for all k ∈ N in order to show that the H-hyponormal matrix T is strongly H-hyponormal. The following result shows that it is sufficient to check this for all k  rank H. Theorem 4.5. Let T ∈ Cn×n be a matrix. If T is strongly H-hyponormal of degree m strongly H-hyponormal.

= rank H, then T is

Proof. It remains to show that (T [∗] )k T k has full domain for all k ∈ N. We will show this by contradiction. Assuming that T is not strongly H-hyponormal, there exists a natural number k  m such that T is strongly H-hyponormal of degree k, but not of degree k + 1. Without loss of generality assume that T and H have the forms as in (7). According to Proposition 4.4 being strongly H-hyponormal of degree k is then equivalent to [∗] i−1 i−1

T2∗ H1 (T1

)

T1

T1

= 0 and T2∗ H1 (T1[∗] )i−1 T1i−1 T2 = 0 for 1  i  k.

We aim to show that (T [∗] )k+1 T k+1 has full domain in contradiction to the assumption that T is not strongly H-hyponormal of degree k + 1. Thus, we have to show [∗] k k

T2∗ H1 (T1

) T1 T1 = 0 and T2∗ H1 (T1[∗] )k T1k T2 = 0.

Note that the size of T1 is m × m, as m

α0 , . . . , αm−1 ∈ C such that (T1[∗] )m =

m −1  i =0

= rank H. Thus, by the Cayley–Hamilton Theorem there exist

αi (T1[∗] )i .

Hence we see that

(T1[∗] )k = (T1[∗] )k−m

−1 m  i =0

(23)

αi (T1[∗] )i .

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M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

This gives m −1  [∗] k k ) T1 T1 = αi T2∗ H1 (T1[∗] )k−m+i T1k T1 i =0 m −1  [∗] k−m+i k−m+i ∗

T2∗ H1 (T1

=

i =0

αi T2 H1 (T1 ) 

!

T1

=0

m −i

T1 T1 "

=0

and [∗] k k )T T

T2∗ H1 (T1

1 2=

=

m −1  i =0 m −1  i =0

αi T2∗ H1 (T1[∗] )k−m+i T1k T2 αi T2∗ H1 (T1[∗] )k−m+i T1k−m+i T1 T1m−i−1 T2 = 0 

!

"

=0

contradicting the assumption. Thus, T is strongly H-hyponormal.



5. Moore–Penrose H-normal matrices In this section we will show, how the sets of H-hyponormal and strongly H-hyponormal matrices are connected to the set of Moore–Penrose-H-normal matrices. Recall that a matrix T ∈ Cn×n is called Moore–Penrose H-normal if HTH † T ∗ H

= T ∗ HT ,

where H † denotes the Moore–Penrose generalized inverse of H. The following lemma can be found in [16, Proposition 4.6]. Lemma 5.1. Let T and H be given as in (7), then the Moore–Penrose generalized inverse of H is given by   −1 H1 0 † H = 0 0 and the matrix T is Moore–Penrose H-normal if and only if    ∗  −1 T1 H 1 T 1 T1∗ H1 T2 H1 T1 H1 T1∗ H1 0 = . T2∗ H1 T1 T2∗ H1 T2 0 0

(24)

Remark 5.2. Note that the equation for the (1, 1)-block means that T1 is H1 -normal with respect to the nondegenerate inner product induced by H1 . Moore–Penrose H-normal matrices were investigated in [11,12]. Each H-normal matrix is Moore– Penrose H-normal, more information is given in the following proposition from [16, Proposition 4.6]. Proposition 5.3. Let T

∈ Cn×n be a matrix. Then the following statements are equivalent:

(i) T is H-normal, i.e., TT [∗] ⊆ T [∗] T ; (ii) T is Moore–Penrose H-normal and T [∗] (T [∗] )[∗]

= (T [∗] )[∗] T [∗] .

In the following example borrowed from [12, Example 6.1], we present a matrix that is Moore– Penrose H-normal, but not H-normal.

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

Example 5.4. Let ⎡ 0 1 H = ⎣1 −1 0 1 then

⎡

H

†

=

3 ⎣4 16 5 1

and furthermore ⎡ ∗

T HT

=

0 ⎣0 0

⎡

⎤ 0 1⎦ , 0 4 0 12 0 0 0

−1

T

⎤ 0 0⎦ 3

0 0 0

=⎣ 0 0

1071

⎤ 5 12⎦ 3 ⎤ 0 0⎦ , 0

⎡ †

∗

HTH T H

=

0 ⎣0 0

0 0 0

⎤ 0 0⎦ . 0

Hence T is Moore–Penrose H-normal. We easily calculate that ⎧⎛ ⎞⎫ ⎨ 1 ⎬ ker H = span ⎝ 0 ⎠ , ⎩ ⎭ −1 which is obviously not T -invariant. Therefore, T is not H-normal. Hence the H-normal matrices are indeed a strict subset of the Moore–Penrose H-normal matrices. The following theorem shows, that in the special case that T is a matrix and T1 in (7) is H1 -normal, the properties Moore–Penrose H-normal, strongly H-hyponormal and H-hyponormal are equivalent. Theorem 5.5. Let T ∈ Cn×n be a matrix and let T and H be in the forms as in (7). Then the following assertions are equivalent: (i) T is Moore–Penrose H-normal; (ii) T is strongly H-hyponormal and T1 is H1 -normal; (iii) T is H-hyponormal and T1 is H1 -normal. Proof. Suppose that (i) holds, i.e., T is Moore–Penrose H-normal. According to Lemma 5.1 this implies [∗]

[∗]

(a) T1 is H1 -normal, i.e., T1 T1 = T1 T1 ; (b) T1∗ H1 T2 = 0, T2∗ H1 T1 = 0 and T2∗ H1 T2

= 0.

Therefore, [∗]

y1∗ H1 (T1 T1

− T1 T1[∗] )y1 = 0 for all y1 ∈ Cm ,

and thus T [∗] T − TT [∗] is H-nonnegative by Proposition 3.3 as T2∗ H1 T2 = 0. Furthermore, due to the H1 -normality of T1 and the fact that T2∗ H1 T1 = 0 and T2∗ H1 T2 = 0, we obtain [∗] k−1 k−1

T2∗ H1 (T1

)

T1

T1

= T2∗ H1 T1k−1 (T1[∗] )k−1 T1 = 0 for k ∈ N

T2

= T2∗ H1 T1k−1 (T1[∗] )k−1 T2 = 0 for k ∈ N.

and [∗] k−1 k−1

T2∗ H1 (T1

)

T1

This proves that (T [∗] )k T k has full domain for all k ∈ N, see Proposition 4.4. Hence T is strongly H-hyponormal. By definition every strongly H-hyponormal matrix is H-hyponormal, therefore (ii) implies (iii). Finally, we will show that (iii) implies (i). Let T be H-hyponormal. Then the fact that T [∗] T has full domain implies that T2∗ H1 T1 = 0 and T2∗ H1 T2 = 0 (see Proposition 2.6), consequently T1∗ H1 T2 = 0.

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Together with the assumption of T1 being H1 -normal and Lemma 5.1 we find that T is Moore–Penrose H-normal.  There are strongly H-hyponormal matrices such that T1 is not H1 -normal, see Example 4.2. Hence, according to Theorem 5.5, the Moore–Penrose H-normal matrices are a strict subset of the strongly Hhyponormal matrices and, hence, of the H-hyponormal matrices (see also Example 3.8). The diagram below shows the relation between the different classes of matrices. Diagram 5.6

6. Invariant maximal nonpositive subspaces of strongly H-hyponormal matrices The question under which conditions invariant semidefinite subspaces can be extended to invariant maximal semidefinite subspaces was discussed in [12–14] for H-normal and H-hyponormal matrices in the nondegenerate case. The fact that in the case of a degenerate inner product ker H remains invariant for an H-normal matrix was the key to a first generalization of those extension results in [16, Theorems 4.7 and 4.8]. We have seen in Example 4.2 that ker H is in general not T -invariant for a strongly H-hyponormal matrix T . The following theorem however describes how to find a T -invariant subspace that contains ker H. Theorem 6.1. Let T ∈ Cn×n be a strongly H-hyponormal matrix. Let M be the smallest T-invariant subspace containing the kernel of H . Then M is H-neutral. In particular, if T and H are in the forms (7), ˙ kerH, where M0 (canonically identified with a subspace of Cm ) is H1 -neutral and the then M = M0 [+] smallest T1 -invariant subspace that contains the range of T2 . Proof. For the proof, we use an idea similar to the one in [12, Theorem 6.6]. Without loss of generality, we may assume that T and H are in the forms (7), where T1 ∈ Cm×m and T2 ∈ Cm×(n−m) . Let X

= [T2 T1 T2 · · · T1m−1 T2 ] and M0 = Im X,

i.e., M0 is the controllable subspace of the pair (T1 , T2 ). Therefore, M0 is the smallest T1 -invariant subspace that contains the range of T2 . In particular, it follows that there exist matrices B and C of appropriate dimensions such that

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

T1 X

1073

= XB and T2 = XC .

Now we identify M0 canonically with a subspace of Cn and set   &:= M0 + ˙ ker H = Im X 0 ⊆ Cn . M 0 I & contains ker H and by Then M     T1 X X 0 T2 = T 0 I T3 X T4





X

0 I

= 0

B T3 X

C T4



& is the smallest T -invariant subspace containing ker H, because & is T -invariant. Moreover, M M   T T (ker H ) = Im 2 T4 and thus any T -invariant subspace containing the kernel of H must also contain M0 . It remains to & = M is H-neutral, or, equivalently, that M0 is H1 -neutral. Thus, let x ∈ M0 . Then there show that M exist xi ∈ C(n−m) and αi ∈ C, i = 1, . . . , m such that x

=

m  i =1

αi T1i−1 T2 xi . [∗] i i ) T1 has full domain for all i, or, by Proposition 4.4 equivalently, [∗] i −1 0 and T ∗ H (T )i−1 T T = 0 for all i, we obtain that

Then using the fact that (T1 [∗] i−1 i−1 ) T T

T2∗ H1 (T1

1

x ∗ H1 x =

=

1

m  i,j=1 m  i,j=1

=

2

1

1

1

2

αi α j xj∗ T2∗ H1 (T1[∗] )j−1 T1i−1 T2 xi $

j −1

αi α j xj∗ T2∗ H1 (T1[∗] )i−1 T1 T2

hence M0 is H1 -neutral.

%∗

xi

= 0,



Finally, we give sufficient conditions for the existence of an invariant maximal H-nonpositive subspace for strongly H-hyponormal matrices by giving conditions when the subspace M from Theorem 6.1 can be extended to a maximal H-nonpositive subspace. Theorem 6.2. Let T be strongly H-hyponormal, and let M be the smallest T-invariant subspace containing the kernel of H which is H-neutral by Theorem 6.1. Decompose M[⊥] as [⊥]

M

˙ Mnd , = M+

(25)

3 the compressions of T and H to Mnd , T33 and H for an H-nondegenerate subspace Mnd . Denote by  3 is invertible. Assume that M is invariant under T [∗] and that, in addition, one of the respectively. Then H three following conditions holds: [∗] σ ( T33 +  T33 ) ⊂ R; [∗] T33 −  T33 ) ⊂ iR; (ii) σ (

(i)

(iii)  T33 is H3 -normal.

Then M can be extended to a maximal H-nonpositive subspace M− that is invariant under T . The conditions (i)–(iii) are independent of the particular choice of a nondegenerate subspace Mnd subject to (25). Proof. The proof is divided into three steps. In the first step, we construct an H-nondegenerated ˙ M3 . In the second step, we show that if one of the conditions (i)– subspace M3 such that M[⊥] = M+

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(iii) for the compression T33 of T to M3 is satisfied, then M can be extended to a maximal H-nonpositive subspace M− that is invariant under T . In the last step, we show that if one of the conditions (i)–(iii) for the compression of T to some nondegenerate subspace Mnd subject to (25) holds, then this condition is also true for T33 . Step (1). Assume that T and H are in the forms (7), with T1 , H1 ∈ Cm×m . Let M0 be as in the proof of Theorem 6.1, i.e., M

˙ ker H . = M0 [+]

As M is H-neutral, the isotropic part M◦ of M equals M. By [18, Lemma 3.10] we find a subspace Msl skewly linked to M0 and a subspace M3 with

˙ Msl )[+] ˙ M3 )[+] ˙ ker H, Cn = ((M0 + [⊥]

(26)

˙ M3 = M0 [+] ˙ M3 [+] ˙ ker H . = M[+]

M

˙ Msl , Note that the space M3 is H-nondegenerate. (This follows from the fact that the space M0 + canonically identified with a subspace of Cm , is H1 -nondegenerate, and, therefore, M3 , also canon˙ Msl . Hence it is ically identified with a subspace of Cm is the H1 -orthogonal companion of M0 + H1 -nondegenerate.) Thus, the restriction H3 of H to M3 is invertible. Step (2). Assume that one of the conditions (i)–(iii) for the compression T33 of T to M3 is satisfied. As M0 (canonically identified with a subspace of Cm ) is H1 -neutral, T1 -invariant, and contains the range of T2 we find, with respect to the decomposition (26) that H and T have the forms ⎤ ⎡ 0 I 0 0   ⎢ H1 I 0 0 0 0⎥ ⎥, H= =⎢ ⎣ 0 0 H 0⎦ 0 0 3 0 0 0 0 and

T

=



T1 T3

T2 T4



⎡

=

T11 ⎢ 0 ⎢ ⎣ 0 T41

T12 T22 T32 T42

T13 T23 T33 T43

⎤ T14 0 ⎥ ⎥. 0 ⎦ T44

[∗]

Then for T1 , we obtain ⎡ ∗ ∗ T22 T12 ∗ ⎢ 0 [∗] T T1 = ⎣ 11 −1 ∗ −1 ∗ H3 T13 H3 T23

∗ H ⎤ T32 3 0 ⎥ ⎦ [∗]

T33

[∗] and as $ %M was assumed to be invariant for T (recall from [16] that by definition this means “x

and

x y

∈

[∗]

T1

T [∗]

=

⇒ y ∈ M”), we obtain that

⎡ ∗ T22 ⎢ 0 ⎣ 0

∗ T12 ∗ T11

−1

∗ H3 T13

∈M

∗ H ⎤ T32 3 0 ⎥ ⎦, [∗]

T33

[∗]

[∗]

or, equivalently, T23 = 0. If we decompose N := H1 (T1 T1 − T1 T1 ) with respect to the decomposition ˙ Msl )[+] ˙ M3 , we obtain that the (3, 3)-block N33 takes the form (M0 + N33

[∗] [∗] = H3 (T33 T33 − T33 T33 ).

Now we show that T33 is H3 -hyponormal. Indeed, let d be the dimension of M3 , let y3 ∈ Cd be arbitrary, and let y1 = (0, 0, y3T )T ∈ Cm . Then T2∗ H1 y1 = 0 and thus, since T is H-hyponormal, we obtain that [∗]

y1∗ H1 (T1 T1

− T1 T1[∗] )y1 = y1∗ Ny1  0.

M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

1075

This implies y3∗ N33 y3  0, and T33 is H3 -hyponormal. It is obvious, that {0} is an H3 -nonpositive, T33 -

invariant subspace, with {0}[⊥]H3 = M3 . As T33 was assumed to be H3 -hyponormal we can use a result from [14, Theorem 6], to obtain that there is a maximal H3 -nonpositive subspace N3 that is invariant [∗]H

under T33 and T33 3 . Canonically identifying N3 with a subspace of Cn , we obtain using the fact that ˙ N3 + ker H is a T -invariant maximal H-nonpositive subspace containing T23 = 0 that M− := M0 + M. Step (3). Next, we show that the conditions (i)–(iii) are independent of the particular choice of a nondegenerate subspace Mnd . Indeed, choosing another nondegenerate subspace Mnd in ⎡ ⎤ I 0 0 ⎢ 0 0 0⎥ [⊥] ⎥ M = Im ⎢ ⎣0 I 0⎦ 0 0 I in place of M3 amounts to a change of basis in Cn given by a matrix of the form ⎡ ⎤ I 0 S13 0 ⎢0 I 0 0⎥ ⎥, S=⎢ ⎣0 0 S33 0⎦ 0 0 S43 I with an invertible S33 . We have ⎡ −1 I 0 −S13 S33 ⎢0 I 0 ⎢ S −1 = ⎢ −1 ⎣0 0 S33 −1 0 0 −S43 S33

⎤ 0 0⎥ ⎥ ⎥. 0⎦ I

Thus, we obtain that with respect to the new decomposition

˙ Msl )+ ˙ Mnd )+ ˙ ker H Cn = ((M0 + and the new basis, the matrices of interest take the form ⎡ ⎤

∗

 T =S

−1

TS

=

⎢0 ⎢ ⎣0

∗

and

⎡  H

= S∗ HS =

0 ⎢I ⎢ ⎣0 0

∗ ∗ ∗ ∗

∗

0 −1 S33 T33 S33

I 0 ∗ S13 0

0 S13 ∗ H S S33 3 33 0

∗

∗

0⎥ ⎥ 0⎦

∗

⎤ 0 0⎥ ⎥. 0⎦ 0

3 of  1 to Mnd are Thus the compressions  T33 and H T1 , respectively, H  T33

−1 3 = S ∗ H3 S33 . = S33 T33 S33 , H 33 −1

Clearly it follows from this that if each of the three conditions (i)–(iii) holds for  T33 = S33 T33 S33 and 3 = S ∗ H3 S33 , then it holds also for T33 and H3 . In particular, the conditions (i)–(iii) are independent H 33 of the choice of Mnd .  Theorem 6.2 generalizes [12, Theorem 7]. Note that already in the nondegenerate case additional assumptions were necessary to guarantee existence of invariant maximal nonpositive subspaces and it was shown in [12] that these assumptions were essential. 7. Conclusions We have extended the notion of H-hyponormality to the case of degenerate indefinite inner products. The straightforward approach to define H-hyponormality, i.e., calling a matrix T H-hyponormal if

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M.-A. Henn et al. / Linear Algebra and its Applications 433 (2010) 1055–1076

H (T [∗] T

− TT [∗] )  0,

turned out not to be satisfactory. Therefore, we developed the new concept of strong hyponormality. The set of strongly hyponormal matrices has the following three useful properties: (i) it contains the sets of H-normal and Moore–Penrose H-normal matrices; (ii) it equals the set of H-normal matrices if H is negative semidefinite; (iii) any strongly H-normal matrix has an invariant H-neutral subspace containing ker H. In particular, we have shown how the latter property can be used to generalize existence results for invariant maximal nonpositive subspaces from the case of nondegenerate indefinite inner products to degenerate ones. References [1] J. Bognár, Indefinite Inner Product Spaces, Springer Verlag, Heidelberg/Berlin/New York, 1974. [2] A. Dijksma, H.S.V. de Snoo, Symmetric and selfadjoint relations in Krein spaces II, Ann. Acad. Sci. Fenn. Math. 12 (1987) 199–216. [3] I. Gohberg, P. Lancaster, L. Rodman, Matrices and Indefinite Scalar Products, Birkhäuser, Basel, 1983. [4] I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra, Birkhäuser, Basel, 2005. [5] I. Gohberg, B. Reichstein, On classification of normal matrices in an indefinite scalar product, Integral Equations Operator Theory 13 (1990) 364–394. [6] R. Grone, C.R. Johnson, E.M. de Sá, H. Wolkowicz, Normal matrices, Linear Algebra Appl. 87 (1987) 213–255. [7] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser, Basel, 2006. [8] M.G. Kre˘ın, Topics in differential and integral equations an operator theory, in: I. Gohberg (Ed.), Oper. Theory Adv. Appl., vol. 7, 1983. [9] P. Lancaster, L. Rodman, Algebraic Riccati Equations, The Clarendon Press/Oxford University Press, New York, 1995. [10] H. Langer, R. Mennicken, C. Tretter, A self-adjoint linear pencil Q − λP of ordinary differential operators, Methods Funct. Anal. Topology 2 (1996) 38–54. [11] C.K. Li, N.K. Tsing, F. Uhlig, Numerical ranges of an operator on an indefinite inner product space, Electron. J. Linear Algebra 1 (1996) 1–17. [12] C. Mehl, A. Ran, L. Rodman, Semidefinite invariant subspaces: degenerate inner products, Oper. Theory Adv. Appl. 149 (2004) 467–486. [13] C. Mehl, A.C.M. Ran, L. Rodman, Hyponormal matrices and semidefinite invariant subspaces in indefinite inner products, Electron. J. Linear Algebra 11 (2004) 192–204. [14] C. Mehl, A.C.M. Ran, L. Rodman, Extension to maximal semidefinite invariant subspaces for hyponormal matrices in indefinite inner products, Linear Algebra Appl. 421 (2007) 110–116. [15] C. Mehl, L. Rodman, Symmetric matrices with respect to sesquilinear forms, Linear Algebra Appl. 349 (2002) 55–75. [16] C. Mehl, C. Trunk, Normal matrices in degenerate indefinite inner product spaces, Oper. Theory Adv. Appl. 175 (2007) 193–209. [17] V. Mehrmann, Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems,Linear Algebra Appl. 121 (1989) 291–331. [18] F. Philipp, C. Trunk, G-Selfadjoint operators in almost Pontryagin spaces, Oper. Theory Adv. Appl. 188 (2009) 207–236. [19] B.C. Ritsner, The Theory of Linear Relations, Voronezh, Dep. VINITI, No. 846-82, 1982 (Russian).